Download Detecting Non-Abelian Anyons by Charging Spectroscopy

PRL 110, 106805 (2013)
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PHYSICAL REVIEW LETTERS
Detecting Non-Abelian Anyons by Charging Spectroscopy
G. Ben-Shach,1 C. R. Laumann,1 I. Neder,2 A. Yacoby,1 and B. I. Halperin1
1
2
Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
Raymond and Beverly Sackler School of Physics and Astronomy, Tel-Aviv University, Tel Aviv 69978, Israel
(Received 5 December 2012; published 8 March 2013)
Observation of non-Abelian statistics for the e=4 quasiparticles in the ¼ 52 fractional quantum Hall
state remains an outstanding experimental problem. The non-Abelian statistics are linked to the presence
of additional low energy states in a system with localized quasiparticles, and, hence, an additional low
temperature entropy. Recent experiments, which detect changes in the number of quasiparticles trapped in
a local potential well as a function of an applied gate voltage, VG , provide a possibility for measuring this
entropy, if carried out over a suitable range of temperatures, T. We present a microscopic model for
quasiparticles in a potential well and study the effects of non-Abelian statistics on the charge stability
diagram in the VG T plane, including broadening at finite temperature. We predict a measurable slope
for the first quasiparticle charging line and an even-odd effect in the diagram, which is a signature of nonAbelian statistics.
DOI: 10.1103/PhysRevLett.110.106805
PACS numbers: 73.43.Cd, 05.30.Pr, 71.10.Pm
The unambiguous observation of particles obeying nonAbelian statistics remains an outstanding experimental
challenge in condensed matter physics. The Moore-Read
fractional quantum Hall state [1], believed to be realized at
filling fraction ¼ 5=2, is one of the most promising
candidate phases to exhibit such quasiparticles (QPs) [2].
The Moore-Read state is predicted to support QPs of charge
e=4; for N such QPs, localized and well separated from
each other, there should be a nearly degenerate set of ground
states, with multiplicity 2N=21 . For temperature T larger
than the splitting of these ground states, but smaller than the
gap to higher excited states, this degeneracy contributes an
effective entropy to the system, the non-Abelian entropy.
Non-Abelian statistics predict that pairs of QPs can
interact to form two distinct states, or fusion channels, f,
commonly denoted as f ¼ 1, c . In a finite system, the two
states have different energies, and the ground state is
unique; for T below the splitting between the two, the
non-Abelian entropy is lost.
There are several recent theoretical proposals for techniques to observe this entropy through bulk measurement
of thermodynamic and transport properties [3–6]. Recent
measurements in this direction of thermoelectric response
at ¼ 5=2 are encouraging [7]. These theoretical proposals assume that all QPs are well separated, such that
degeneracy-lifting interactions are weak or nonexistent.
However, recent local electronic charge-sensing measurements, using a single-electron transistor (SET) [8], suggest
that QPs tend to trap in local potential wells due to electrostatic disorder, which may be tightly confining and contain
more than one QP. In this experiment, by comparing the
spacing between charging events at ¼ 5=2 with those at
7=3, the QP charge was verified to be e=4 as predicted.
Confined QPs split their degeneracy through two means:
Majorana exchange [9,10] present even for stationary QPs
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and as we show here, an orbital splitting from interchange
of the charged QPs, which can dominate in special cases.
In this Letter, we study the charging spectra of local
quasiparticle traps. Such traps may be induced by disorder
or defined by gates. Their spectra reflect the QP statistics,
just as electronic dot spectra reflect the spin and fermionic
statistics of electrons. We show that low frequency SET
charge-sensing measurements, which provide only
thermally-averaged information regarding the dot spectra,
are sufficient for extracting non-Abelian signatures. At low
but experimentally accessible T, we predict a robust temperature evolution of the N ¼ 0–1 transition, and an evenodd effect in the evolution of the charging spectrum for
several non-Abelian anyons. This effect should be visible
for T below the relevant gaps to excited states for N
particles, which we calculate for N ¼ 1, 2.
The experiments of Venkatachalam et al. [8] measure
the change in potential at the SET induced by a change
VG in the potential applied to a backgate on the sample. If
there is a single disorder-induced well close to the SET, the
measured signal is inversely proportional to the compressibility of the well, ¼ @hNi
@ , where is the QP chemical
potential in the vicinity of the well. For an isolated well, the
relation between VG and the change in should be linear,
but the constant of proportionality is geometry-dependent,
as screening depends on the local environment as well as
the distance to the gate [11]. If there are several wells
nearby, their signals are weighted according to the strength
of their coupling to the SET; in this case, Coulomb interactions between wells also need to be taken into account.
At T ¼ 0, the compressibility has a -function peak at
a crossing of energy levels between N and N þ 1 QPs in a
well. At finite T, the peak broadens and may shift as a
function of due to entropy effects. The simplest case to
consider is an isolated well at the transition from N ¼ 0 to
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N ¼ 1, or slightly more involved, from one to two. At
higher occupation numbers, we give qualitative arguments
for the stability diagram. We examine both circular and
elliptical traps, and account for temperature effects including broadening and excited states. In our model, e=4 QPs
are represented as interacting charged particles in a magnetic field, confined to the lowest Landau level (LLL), with
non-Abelian statistics. The interaction is Coulomb, supplemented by an interaction VX ðrÞ due to the exchange of
Majorana fermions.
Qualitative Picture.—We begin with the charging diagram for Abelian particles in a well to contrast it with the
non-Abelian case. For simplicity of presentation, we consider varying only the QP chemical potential, although as
discussed below, local gating will be required to access the
full charging spectrum. The well sits in a larger quantum
Hall state containing other distant wells, which provide a
reservoir for QPs. At T ¼ 0, as a function of chemical
potential, a series of peaks in the compressibility appear,
corresponding to individual charging events in the well.
The spacing of these peaks defines the charging energy,
UðNÞ. As T increases, the peak centers evolve vertically in
the charging diagram (red, dashed lines in Fig. 1), until T
reaches the minimum excitation energy, N , set by the
excited states within the well. Above this energy, the
curve deviates from a straight line due to entropic effects.
The peaks broaden linearly with T for both Abelian and
non-Abelian QPs.
When several non-Abelian QPs occupy a tightly confining well, they uniquely fuse at low energies. This produces
a distinct experimental signature-the even-odd effect. As
highlighted in Ref. [4], the density dependence of the zerotemperature entropy produces a distinct signature in the
inverse compressibility of bulk samples at low T. In local
traps with discrete QP number, the difference in zerotemperature entropy S between adjacent number states
produces a related low-T signature in the charge stability
FIG. 1 (color online). Cartoon charge stability diagram, showing only peak centers (no broadening). Vertical axis is temperature T; horizontal axis is the chemical potential for charged
QPs, controlled in experiment by a gate potential. Red, dashed
lines are for Abelian particles. Blue, solid lines correspond to
non-Abelian QPs in a tightly confining well. N is the gap to
excited states for N particles and sets scale for other entropic
effects. Notice even-odd effect for non-Abelian anyons.
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diagram: the slope of the charge transition line in the
T plane is 1=S. The first QP placed in the well
contributes SNA ¼ ln2=2 to the
pffiffiffi non-Abelian entropy
(kB ¼ 1), or equivalently adds a 2 degeneracy; thus, the
N ¼ 0–1 transition line has slope 2= ln2 in the T
plane as T ! 0. A second QP fuses uniquely with the first
QP into the 1 or c channel and the non-Abelian entropy is
extinguished, S ¼ ln2=2. Thus, as T increases from
zero, the N ¼ 1 state becomes entropically more favorable
than the N ¼ 2 state, and the transition line has slope
þ2= ln2 (blue, solid lines in Fig. 1). This even-odd effect
persists as the well charges: odd numbers of particles fuse
into the non-Abelian -channel, while even numbers
uniquely fuse into either the Abelian 1 or c channels, as
long as T remains below the splitting between these two
channels. If the splitting between channels is smaller than
N , there exists an intermediate regime, in which the
degeneracy is preserved and the non-Abelian entropy
increases by ln2=2 with every additional particle, and all
lines have parallel negative slopes. A similar effect for
electrons due to spin degeneracy was predicted and seen
in quantum dots at B ¼ 0 [12,13]. For a quantized Hall
state in a strong magnetic field, however, if QPs have more
than one spin state, we expect their energies to split, due to
the Zeeman and/or Coulomb exchange fields, by an
amount that is large compared to the temperatures of
interest. Moreover, if spin-degeneracy were present, the
charging lines would have a different slope than for fusion
channel degeneracy because of the factor of two difference
in entropy per QP.
For wells far apart compared to the magnetic length, the
rate, well =@ of Majorana exchange between them falls off
exponentially in their separation. We, therefore, consider
charging lines for temperatures T well , assumed zero
for an isolated well. In this limit there is no fusion-channel
splitting between wells and each independently exhibits
the even-odd effect. However, the charging spectra are not
completely independent due to capacitive coupling. In an
experiment sensitive to multiple disorder-induced wells,
the charging spectra of the wells appear overlaid with
unknown offsets making the even-odd effect more difficult
to observe, without first associating the various peaks to
their respective wells. Experiments [8,14,15] suggest that
determining such associations is possible.
Equilibration.—Although QPs are locally trapped, the
equilibrium model we present requires that the system
explores the degenerate ground-state manifold faster than
the measurement time texp of the charge-sensing experiments. This time is determined by the rate of change of the
gate voltage, typically texp 0:1 s. We estimate the equilibration time due to thermal excitations as tT 104 s texp , meeting the requirement [11]. Moreover, the observed
changes in the charge state of the studied well during
experiments [8] imply that QPs hop freely between wells
on the time scale texp . Assuming the hopping processes
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have a stochastic component, they will naturally lead to
braiding of QPs from different wells.
Quantitative Picture.—Returning to a single well in a
large bath, we present a model for calculating the charging
diagram. The partition function is
X
Z ¼ gðNÞeðFðN;TÞNÞ ;
(1)
N
where ¼ 1=T, the internal free energy of N-particle
states in the well is FðN; TÞ, and
gðNÞ ¼
8 pffiffiffi
< 2
:1
N odd
(2)
N even
captures the non-Abelian degeneracy associated with net
fusion within the well. In a well where QPs are close, such
that all other fusion-degeneracies are split by energies
larger than T, we take FðN; TÞ FðN; 0Þ for T N ,
the gap to excitations. In principle, however, for a wide
well where electron-electron interactions localize the QPs
further apart, an intermediate regime can exist in which the
topological degeneracies are not significantly split and
FðN; TÞ FðN; 0Þ TbN2 c ln2, where bc denotes the integer part, for temperatures up to the gap N .
The compressibility follows from the partition function.
To leading order near the N 1 to N charge transition at the
critical chemical potential, N
0 FðN; 0Þ FðN 1; 0Þ,
¼
ð1
gðNÞ
ðFÞ
gðN1Þ e
gðNÞ
þ gðN1Þ
eðFÞ Þ2
;
(3)
where ¼ N
0 and F ¼ FðN;TÞ FðN 1;TÞ .
We
differentiate
with respect to to find the center
N
0
of the peak: max ¼ T lnðgðN 1Þ=gðNÞÞ þ F. For a
tightly confining circular well at low T, for which F ¼ 0,
this gives max ¼ ðT=2Þ ln2, which confirms that the
charging line slopes alternate sign as a function of the parity
of N. In the intermediate regime, the slope is negative for all
N. The peak height decreases with T as max 1=4T, while
the full-width-half-max (FWHM) increases with
pffiffiffi T due to
number fluctuations as FWHM 2T lnð3 þ 2 2Þ, roughly
ten times as fast as the shift in position. Nevertheless,
tracing the peak should be possible if measurements are
sufficiently accurate. In the experimental regime of interest,
N1
the charging energy UðNÞ ¼ N
T, so the
0 0
peaks remain distinguishable. The key input to the above
statistical model is the microcanonical low-energy spectra
of fixed numbers of QPs in a well, which we now calculate
for N ¼ 1, 2.
One Particle.—For a particle in an elliptical harmonic
well,
1
Vtrap ¼ kðx2 þ y2 Þ;
2
(4)
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where k is the spring constant, and controls the eccentricity ( ¼ 1 defines a circular trap), the level spacing is
pffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffi
1 ¼ k l2
B , where lB ¼ @=e B is the effective magnetic length for QPs in a magnetic field B. At finite T, this
produces an internal free energy,
Fð1; TÞ ¼ T lnð1 e1 =T Þ:
(5)
This free energy decreases weakly with T for T < 1 , only
significantly correcting the linear charging curve for
T 1 , as shown in Fig. 1.
Two Particles.—As the fusion channel, f, of two orbiting
non-Abelian anyons is conserved, the orbital dynamics
may be treated separately in each f-sector. This reduces
to the dynamics of Abelian anyons whose statistical angle
depends on the fusion sector. For Ising anyons, 1 ¼ 0
and c ¼ =2 [16]. To model two such anyons in a well,
each with charge e ¼ e=4, we write the Hamiltonian for a
pair of bosons with a statistical gauge field as
H¼
2
1 X
ðp~ @a~ fi e A~ i Þ2 þ Vtrap ðr~i Þ þ VI ðr~1 r~2 Þ
2m i¼1 i
þ VXf ð~r1 r~2 Þ:
(6)
The first term contains the electromagnetic vector potential
A~ i , corresponding to a uniform external B-field, as well as
a statistical gauge field ðafx ; afy Þ ¼ rf2 ðy; xÞ, which binds
a flux tube of strength f to each quasiparticle, and m is the
effective QP mass. We project into the LLL, taking m ! 0.
The coordinates in a~ i are relative to the other particle. We
assume that the QPs interact via a Coulomb interaction,
e2
VI ¼ 4
r
, where r 0 is the electric permittivity of
the material. This approximation is valid assuming that
QPs do not come within lB of each other. VX is the direct
energy splitting of the fusion channels due to virtual exchange of Majorana fermions. It is related to the fusion
channel splitting discussed in Refs. [9,10] and should
consist of an exponential decay and oscillations, each on
the order of several lB . For circular wells, the behavior of
Abelian anyons has been treated previously [17–20]. We
summarize key results and include corrections due to
eccentricity.
In the symmetric gauge for harmonic traps, the centerof-mass (c.m.) and relative (REL) coordinates decouple. In
the c.m. coordinate, the statistical gauge field a~ f falls
out, leaving a single particle projected into the LLL in a
harmonic well. For the REL coordinate, the particle is
confined to a half-plane with the origin removed, and
a~ f remains [18]. We change the gauge, so that a~ f ¼ 0,
giving a twisted boundary condition, c REL ðr; Þ ¼
eif c REL ðr; 0Þ. The potential landscape in the half-plane
is defined by strong Coulomb repulsion near the origin
together with the harmonic trap, Vtrap þ VI , for VX ¼ 0.
The twisted periodic boundary conditions allow
only angular momenta ‘ ¼ 2n þ =, for n integer.
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The REL-coordinate wave functions in the LLL have a
basis given by j‘i,
hzj‘i ¼ N‘ð1=2Þ z‘ ejzj
2 =4ð2l2 Þ
B
;
(7)
where z ¼ x þ iy and N‘ is a normalization constant on the
half-plane. In this basis, we can diagonalize to find the twoparticle spectrum. The potential has diagonal terms, as well
as an off-diagonal term only when circular symmetry is
broken [11].
Circular Well.—We assume VX ¼ 0 initially, and note
that the c.m. coordinate behaves just like the single particle
case with c:m: ¼ 1 . The lowest energy gap fR in the
relative coordinate within a fusion channel f can be found
by taking differences between adjacent ‘-states near the
minimum, obtained by diagonalizing the Hamiltonian. We
define the parameter r0 ¼ ðe2 =2
kÞ1=3 , the radial position of the minimum of the potential. This splitting fR
oscillates with r0 at fixed magnetic field with an amplitude
that decays in the large well limit, r0 lB , as
fR & 121
r20
e2 l4
B
¼
24
:
4
r50
l2
B
(8)
The other relevant gap for the relative coordinate is the
energy difference E1 c ¼ jE10 E0c j between lowest
energy states in the 1 and c channels. With VX ¼ 0, the
splitting between fusion channels is an interchange effect,
which follows from the allowed angular momenta in each
channel; in particular, E1c behaves similarly to fR with a
4
9 e2 lB
2 4
r50 ,
of fR . For
maximum oscillation bounded by the power law
which is approximately 20% of the amplitude
T < E1 c and fR , the slope of the 1–2 transition in the
T plane is positive, exhibiting the even-odd effect.
Clearly, intra-channel entropy washes out the effect for
T > fR . As r0 varies, E1 c will oscillate in sign, and can
be arbitrarily small if r0 is close to a zero-crossing. If
E1 c < T < fR , the 0–1 and 1–2 charging lines are parallel
with negative slope 2= ln2 [11].
Non-Abelian QPs at finite separation can exchange
Majorana fermions, leading to an additional fusion channel
splitting. Unlike the orbital contribution, this splitting
occurs even when QPs are localized. Using a variational
method to calculate this energy splitting for particles on a
sphere, it was found to decay exponentially on the order of
several magnetic lengths, up to a numerical prefactor of
Oð1Þ [10]. VX in the Hamiltonian accounts for a splitting
of this form. We do not calculate VX explicitly, but note
that while it dominates the shift between fusion channel
spectra in tightly confining wells, it oscillates and decays
exponentially as the well widens and particle separation
increases. In general, VX increases E1 c , promoting the
even-odd effect over a larger T-range, and making a regime
of parallel charging lines less likely.
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Anisotropic Well.—For anisotropic wells, again taking
VX ¼ 0 initially, consider the relative coordinate for two
QPs. Starting from the circular well where QP orbits
encircle the origin in the half-plane, as the eccentricity increases, the effective potential acquires a minimum on
the x-axis, at x ¼ r0 , and a saddle point on the y-axis at
y ¼ r0 =1=3 . For any given > 1, the wave function
becomes effectivelypffiffiffiffiffiffiffi
localized
near the potential minimum
ffi
1
ðÞ.
This is when the lowest
for ðr0 =lB Þ2 > p2ffiffi3 ð1=3
1Þ
energy state near the minimum has energy lower than the
saddle point potential. As ! 1, ðÞ diverges as
ð 1Þ1=2 , confirming that for a circular well QPs are
not localized. Eccentricity breaks any accidental degeneracies which arise in the circular potential near r0 , and
modifies the spectrum of the well. For low eccentricities,
the degeneracy breaking can increase or decrease the orbital
splitting. For large enough , the QPs are trapped at opposite ends of the well and no longer orbit each other except
for quantum tunneling across the saddle point. In a saddle
point tunneling model, the orbital exchange rate, R, in the
large well limit is Gaussian in the well size, R 1=2
ðÞ1 cðÞðr0 =lB Þ2 , where cðÞ depends
kl2
B exp½
weakly on and goes to a constant of order unity as ! 1
[11]. This expression may be obtained by estimating the
potential as Harmonic near the minimum, and using a
WKB-type calculation of the tunneling of a particle near a
quadratic saddle point in the LLL, as in Ref. [21]. Increasing
also has the effect of raising the energy of the ground state
by increasing the harmonic frequency of the trap.
For anisotropic wells with VX 0, the exchange effect
naturally dominates the splitting at large r0 , since the
exchange of neutral Majorana fermions decays exponentially while the interchange of localized charged particles
in a magnetic field decays as a Gaussian. We recover the
even-odd effect for T below this splitting, regardless of QP
localization.
Energy Estimates.—A simple model producing a charge
trap is provided by considering a pointlike gate, a distance
d above the two dimensional electron gas. A charge þjej
on this gate produces an effective circular harmonic trap in
jee j
the plane with spring constant k ¼ 4
d
3 . Using r ¼ 13
for GaAs=AlGaAs quantum wells, B ¼ 3:5 T and d ¼
100 nm, we find r0 ¼ 63 nm. The charging energy is
1.6 K, and the gap to single particle excited states in the
well is 1 0:24 K, preserving the slope of 2= ln2
throughout the accessible experimental range 20 mK &
T & 80 mK. The 1-channel ground state has lower energy
than the c -channel by E1 c 29 mK in the absence of VX ,
and the intra-channel gap 1R 220 mK, above the accessible range. As r0 =lB 2:3, we expect the contribution of
VX to enhance the even-odd effect. Since the calculated
charging energy is larger than the energy gap for the ¼
5=2 plateau, it is probably impossible to observe multiple
transitions in a single well simply by changing the voltage
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on a backgate. However, applying a voltage to a pointlike
gate on top of the sample can change the depth of a well by
a large amount without inducing QPs in the surrounding
5=2-state.
To further enhance the even-odd effect, all energy gaps
need to be increased. Increasing the charge on a point gate
or reducing the setback distance d makes the confining trap
tighter. Increasing the magnetic length—by lowering B
while maintaining the filling fraction—increases all of
the relevant splittings in a fixed trap geometry.
Conclusion.—The detection of non-Abelian QPs
through local charge-sensing measurements falls within
realistic experimental parameters. A sensitive compressibility measurement could extract slopes reflecting the
degeneracies of the ground state. Additional control over
confinement potentials will allow for even more conclusive
experiments.
We thank B. Feldman, G. Gervais, S. Hart, V. Lahtinen,
A. Stern, and V. Ventakachalam for useful discussions.
G. B. supported by NSERC and FQRNT. C. R. L. supported by a Lawrence Gollub Fellowship and the NSF
through a grant for ITAMP at Harvard University. I. N.
supported by the Tel Aviv University Center for
Nanoscience and Nanotechnology. This work was supported in part by a grant from the Microsoft Corporation,
and by NSF Grants No. DMR-0906475 and No. DMR1206016.
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